Equational problems anddisunification

Abstract

Roughly speaking, an equational problem is a first order formula whose only predicate symbol is =. We propose some rules for the transformation of equational problems and study their correctness in various models. Then, we give completeness results with respect to some "simple" problems called solved forms. Such completeness results still hold when adding some control which moreover ensures termination. The termination proofs are given for a "weak" control and thus hold for the (large) class of algorithms obtained by restricting the scope of the rules. Finally, it must be noted that a by-product of our method is a decision procedure for the validity in the Herbrand Universe of any first order formula with the only predicate symbol =. 2 F will be assumed to be finite throughout this paper. The case where F is infinite is studied in Lassez, Maher & Marriott (1986) and Maher (1988), it seems to be simpler. 3 This is not the standard definition (as e.g. in Huet & Oppen 1980), but this allows substitutions in any F-algebra A, including the cases where A does not contain X

Anti-pattern Matching

“…In this section we provide a way of transforming any anti-pattern matching problem into a corresponding equational one that has the same set of solutions. We extend the notion of an equation between terms [11] Given an anti-pattern q and a ground term t, we consider the following rewrite system AP-Elim. This transforms an anti-pattern matching problem into an equational one:…”

“…A natural way to solve this type of problem is to use a disunification algorithm such as described in [11]. Due to lack of space, we cannot present disunification in detail.…”

“…Instead we give in Figure 1 the set of rules we consider. The interested reader can refer to [11] for a detailed presentation of disunification.…”

“…In [11] there is a clear separation between the elimination of parameters and the rules that reach definitions with constraints. But, as affirmed both in [11] and [9], such a strict control is only for presentation purposes.…”

“…Of course, many frameworks and results have already contributed to the use of negation in logic based languages. Having in particular in mind negation by failure in Prolog [8], the explicit use of counter-examples [16], disunification [11], feature constraints [3], inclusion constraints [20] and negation in iRho [17], we will motivate and explain the usefulness of anti-patterns.…”